Data points inside the triangle

The triangle described in Section 3.3 provides rigorous bounds on mechanical properties of porous media. For plots in the ($\rho/\mu$, $\lambda/\mu$)-plane such as those included in Figure 1(d) and Figures 3(b), 3(d), and 3(f), some data points lie between the ideal patchy saturation line and the Gassmann ideal lower bound. The relative position of the data points may contain information about the fluid distribution. Consider the case of a core sample that is nearly saturated, above 90% for example. If the weight of the core is used to determine the saturation but the core contains a few gas bubbles, the background saturation will be underestimated and the bubbles themselves represent patches. This is an example of a material having a few isolated patches contained in an otherwise homogeneous partially-saturated background. Such data would plot above but close to the Gassmann curve. In an analogous case for field seismic data, the background saturation may be known from measurements made at lower frequencies or in a nearby region, and it may be possible to use such information to determine the relative volume of patches. For data lying in the middle (i.e., between the bounding curves), some assumptions about fluid distribution could be made and then various estimates about patchy volumes could be applied to different models such as the Hashin-Shtrikman bounds (Hashin and Shtrikman, 1962) or effective medium theories. Exploration of these issues will be the subject of future work.